When reading the history of Erfurt on Wikipedia (in German there is an excellent page on the history of the city) we learn that from 1664 untin 1802 it was part of the electorate of Mainz, after which it went to Prussia. So any Archbishop-elector of Mainz in this period would als be the legitemate ruler of Erfurt.
As the arms of Erfurt and Mainz are the same, there is no simple way to see the difference between the coins that were minted in Mainz or in Erfurt, but my guess is that mintmaster insignia are crucial here (they seem to be totally different from the personal in Mainz, which makes sence given the 300 km between them). Quite often the larger coins of Mainz show more elaborate arms, but the small coins seem very similar.
Apperently the mintage of coins in Erfurt was discontinued until 1756 and then restarted, probably reflecting the growing importance of Erfurt as a center of commerce (today it is the capital of the state of Thüringen), the palace of the Statthalter (sort of governer who ruled the area in name of the bishop) that was build in the later part of the 18th century, was so grand that it housed a conference for German rulers in 1808 and it also served Napoleon as a residence during his campaigns in the area.